L(s) = 1 | − 2·2-s + (3.65 + 3.69i)3-s + 4·4-s + 2.23i·5-s + (−7.31 − 7.38i)6-s + 2.58·7-s − 8·8-s + (−0.280 + 26.9i)9-s − 4.46i·10-s − 43.0·11-s + (14.6 + 14.7i)12-s − 25.4i·13-s − 5.17·14-s + (−8.23 + 8.15i)15-s + 16·16-s + 0.642i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.703 + 0.710i)3-s + 0.5·4-s + 0.199i·5-s + (−0.497 − 0.502i)6-s + 0.139·7-s − 0.353·8-s + (−0.0103 + 0.999i)9-s − 0.141i·10-s − 1.18·11-s + (0.351 + 0.355i)12-s − 0.543i·13-s − 0.0988·14-s + (−0.141 + 0.140i)15-s + 0.250·16-s + 0.00916i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3561079451\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3561079451\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + (-3.65 - 3.69i)T \) |
| 59 | \( 1 + (197. - 407. i)T \) |
good | 5 | \( 1 - 2.23iT - 125T^{2} \) |
| 7 | \( 1 - 2.58T + 343T^{2} \) |
| 11 | \( 1 + 43.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 25.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 0.642iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 132.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 166. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 140. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 338. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 296. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 207. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 54.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 27.5iT - 1.48e5T^{2} \) |
| 61 | \( 1 + 474. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 525. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 977. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 707. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 629.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 776.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 169. iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.02229898482421563875151449639, −10.48863413019382714062406119164, −9.832894329510891902918526995710, −8.572702972869983723530507515636, −8.192047957461301265779014246349, −7.09392525110712839372869642439, −5.70460494884622989784861199142, −4.49927128933195292933942358371, −3.11270017990151191632789592800, −2.08460289956029222804294696182,
0.13018389474859530231036781892, 1.77024578326442371595412513831, 2.74842381553887207252282339638, 4.33244632126467878148831626904, 5.96837653650615583838238103764, 6.91266151630320311079300622824, 7.979473881876616825338566264961, 8.449886665205043347265629071623, 9.450075859610362739701072329665, 10.41109558072012330597820473200